Finding the parts costs the same whether there are any or not

Domineering boards of 4 squares by 5 with different squares blocked out, and what the decomposition is worth on each. The pass that finds the regions is a flood fill and visits every square once, so it costs the same on all of them. What it buys ranges from nothing — on the boards that do not decompose — to a saving of 2,808 positions, and it cannot tell which case it is in until it has run.

1 essay calls finding-cost. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.